A description/proof of that open subspace of locally compact Hausdorff topological space is locally compact
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of locally compact topological space.
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of subspace topology.
- The reader admits the proposition that for any topological space, any subspace subset that is compact on the base space is compact on the subspace.
- The reader admits the proposition that any subspace of any Hausdorff topological space is Hausdorff.
- The reader admits the proposition that any Hausdorff topological space is locally compact if and only if there is a compact neighborhood around any point.
Target Context
- The reader will have a description and a proof of the proposition that any open subspace of any locally compact Hausdorff topological space is locally compact.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any locally compact Hausdorff topological space, \(T_1\), any open subspace, \(T_2 \subseteq T_1\), is locally compact.
2: Proof
For any point, \(p \in T_2\), \(T_2\) is a neighborhood of \(p\) on \(T_1\). There is a compact neighborhood of \(p\), \(N'_p \subseteq T_1\), on \(T_1\) such that \(N'_p \subseteq T_2\) by the definition of locally compact topological space. \(N'_p\) is compact also on \(T_2\), by the proposition that for any topological space, any subspace subset that is compact on the base space is compact on the subspace.
\(T_2\) is locally compact, the proposition that any subspace of any Hausdorff topological space is Hausdorff and the proposition that any Hausdorff topological space is locally compact if and only if there is a compact neighborhood around any point.
3: Note
Note that \(T_1\)'s being Hausdorff is used here, while any closed subspace is locally compact without \(T_1\)'s being Hausdorff.
